Scattering by a Symmetrical Eckart Potential

The Schrödinger equation for scattering of a monoenergetic beam of particles of mass from a symmetric Eckart potential can be written , where is the potential height, is a measure of its width, and is the particle momentum. The equation can be solved exactly in terms of Gauss hypergeometric functions, with details given in the cited reference. In contrast to a classical scattering problem, particles have a finite probability of penetrating the barrier even if their kinetic energy is less than the barrier height—this is an instance of the quantum-mechanical tunnel effect. For an incident beam of unit intensity , the transmitted and reflected beams have intensities and , respectively. The tunneling probability decreases with increasing barrier height and width and drops precipitously for more massive incident particles. Tunneling increases with particle energy, however. Another feature that contrasts with classical behavior is the partial reflection of the wave, even for kinetic energies greater than the barrier height ().
The wavenumber is determined by the mass and energy of the incident particle by . In this Demonstration, units based on are used. The top figure shows the potential barrier and the particle kinetic energy as a dashed horizontal line. The black, blue, and red arrows are labeled with the magnitudes of the incident, transmitted, and reflected waves, respectively. The lower figure shows a plot of the real and imaginary parts of the wavefunction. The amplitudes to the left and right of the barrier are closely related to the three scattering components.


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Snapshot 1: decrease of transmission probability with increasing barrier width
Snapshot 2: decrease with increasing mass
Snapshot 3: partial reflection for
Reference: D. ter Haar, Problems in Quantum Mechanics, London: Pion Ltd., 1975, pp. 11–12, 139–141. There is a misprint in equation (1), where the argument of the second hypergeometric function should read .
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